How Compatible is Perfect Competition with
Transmission Loss Allocation Methods?
Jing Dai, Yannick Phulpin, Vincent Rious, Damien Ernst
To cite this version:
Jing Dai, Yannick Phulpin, Vincent Rious, Damien Ernst. How Compatible is Perfect Competition with Transmission Loss Allocation Methods?. European Electricity Market 2008, May
2008, Lisbonne, Portugal. 6 p., 2008. <hal-00300388>
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How Compatible is Perfect Competition with
Transmission Loss Allocation Methods?
Jing Dai∗ , Yannick Phulpin∗ , Vincent Rious∗ , Damien Ernst†
∗
SUPELEC
Paris, France
E-mail: jing.dai, yannick.phulpin, [email protected]
†
University of Liège
Liège, Belgium
E-mail: [email protected]
Index Terms—Transmission loss allocation, agent-based simulation, market efficiency, electricity market
Abstract—This paper addresses the problem of transmission
loss allocation in a power system where the generators, the
demands and the system operator are independent. We suppose
that the transmission losses are exclusively charged to the
generators, which are willing to adopt a perfectly competitive
behavior. In this context, their offers must reflect their production
costs and their transmission loss costs, the latter being unknown
beforehand and having to be predicted. We assume in this
paper that the generators predict their loss costs from the past
observations by using a weighted average of their past allocated
costs. Under those assumptions, we simulate the market dynamics
for different types of transmission loss allocation methods. The
results show that the transmission loss allocation scheme can lead
to a poorly efficient market in terms of social welfare.
I. I NTRODUCTION
I
N power systems where generation and transmission are
unbundled, the transmission operational costs (e.g. transmission losses or congestion costs) are generally supported by
the System Operator (SO), which charges the generators and
the demands for those costs. Consequently, the transmission
costs are allocated ex post, when the generation and demand
dispatch is known [1]. In this context, allocating transmission
losses has become necessary in order to define locational economic incentives towards a more rational use of ressources [2].
A possible solution could be to allocate active power losses
to the market participants depending on the amount of transmission losses resulting from their injection. However, since
active power losses are a non-separable nonlinear function of
the bus power injections, there is no unique solution to assess
each market participant’s contribution in the transmission
losses [3]. As emphasized in [1] and [4], alternative strategies
have therefore been proposed to design Transmission Loss
Allocation (TLA) methods leading to appropriate economic
signals.
Generally speaking, the efficiency of those economic signals depends on the competition strategies of the market
participants [5], [6]. The most common types of strategies
considered to analyze electricity markets are those related
to the competition models derived from game theory [7]
such as, for example, the Cournot, Bertrand or Stackelberg
models. Besides game theory, agent-based approaches are
also used to analyze electricity markets. Those approaches
model the market as a dynamic system of interacting agents.
An agent refers in this context to a bundle of data and
behaviorial methods representing an entity constituting part
of the simulated market [8]. With respect to the models cited
above (Cournot, Stackelberg), such approaches can provide a
detailed modeling of the market, able to highlight phenomena
that Nash-equilibria types of techniques can not. As way of
example, they are advocated to study the dynamics of the
market before the participants eventually settle to a Nashequilibrium.
The goal of this paper is mainly to study the influence
that TLA methods may have on electricity markets where
the generators are assumed to have a perfectly competitive
behavior. An agent-based approach is chosen to carry out
the study. Each generator is actually modeled as an agent
which formulates offers that reflect its generation cost and
its transmission loss cost. Since individual lost costs are not
known beforehand, each generator must predict those costs.
The prediction of the loss costs may be a difficult task as the
final allocation depends on the predictions of the others. While
utilities could rely on sophisticated approaches to predict the
loss costs, the predictions are assumed, in this paper, to be
carried out by computing a weighted average of the past
allocated loss costs.
The simulation process is summarized in Figure 1. First,
the generators predict their loss costs based on the past
allocation results. Then, their offers are computed by summing
their production costs and their predicted transmission loss
costs. After computing the merit order using those offers, the
different transmission loss costs are sent to the generators and
the overall process repeats. The results of the simulations are
analyzed to determine the efficiency of three TLA methods
(namely the pro-rata, proportional sharing and equivalent
bilateral exchange methods), which is associated here to the
social welfare of the system. This study is carried out on a
Generators
PL×MCP
Loss cost estimation
PL
Estimated loss costs
(PD+PL )×MCP
Bid/offer formulation
Loss costs
Bids/offers
PD
PL×MCP
Market
SO
Market clearing
Generation/demand dispatch
PD×MCP
Loss allocation
Demands
Fig. 1. Simulation scheme for the analysis of the interaction between the
TLA mechanism and the generators’ decisions.
Fig. 2. Power (dashed arrows) and economic flows (continuous arrows) in
the power system.
2-bus system where, in order to simplify the approach, the
demand is fixed and the losses are exclusively allocated to the
generators.
The paper is organized as follows. In Section II, the experimental market design is detailed and a review of some
common TLA approaches is provided. Then, Section III
describes the prediction algorithm. Some criteria to evaluate
the efficiency of TLA methods are presented in Section IV.
Section V presents the results for different TLA methods and a
discussion of their relative performance. And, finally, Section
V concludes.
Cost
Estimated loss cost
PGi
Offer curve
II. M ARKET DESIGN WITH TLA
In this paper, we focus on a specific market design, which
is appropriate to implement different types of TLA methods.
While losses could be considered in the merit order using
locational marginal prices [9], this market design is based
on a lossless economic dispatch. That means that the offers
are selected on their price regardless of their location in the
transmission grid.
In order to ease the approach, we have chosen to allocate
transmission losses exclusively to the generators, which are
usually considered as being the market participants with the
highest price responsiveness. In addition, we consider that the
demands are non-responsive to price changes, i.e. the amount
of power demand is fixed.
In this context, we use a pool-based market, with no bilateral
exchange contract. The inherent physical and economic flows
are represented in Figure 2. Active demands (PD ) and losses
(PL ) are paid at the Market Clearing Price (MCP) and the
SO charges the generators for the transmission loss costs.
We suppose that other transmission costs (such as investment
costs, for example) are supported by the demands only, and
that they are considered in the demand curve.
A. Offer formulation
We suppose that the NG generators are willing to adopt
a perfectly competitive strategy. This assumption means that
the generators bid at their expected marginal cost. In the
context of transmission loss allocation, the expected marginal
cost encompasses the production cost and the expected loss
Marginal production cost
Fig. 3. Offer curve and marginal production costs of a generator i as a
function of its active power generation PGi .
cost. Every generator i is supposed to know perfectly well
P
its production cost CG
(PGi ) and, in addition, it estimates
i
L
its loss cost ĈGi (PGi ), which is to be integrated in its offer
O
P
L
ĈG
(PGi ) = CG
(PGi ) + ĈG
(PGi ). Figure 3 plots a typical
i
i
i
offer curve of a generator i.
B. Merit order
The process of merit order determines the generation dispatch PGi , . . . , PGNG and the market clearing price (M CP )
O
O
(PGNG ). It is com(PG1 ), . . . , ĈG
based on the offers ĈG
NG
1
puted using an optimal power flow formulation as detailed
below.
min
PG1 ,...,PGN
O
O
(PGNG ))
(PG1 ), . . . , ĈG
M CP (ĈG
N
1
G
(1)
G
subject to:
NG
X
i=1
P Gi = P L +
ND
X
PDj
(2)
j=1
and the other classical load-flow equations [10].
Equation (2) and the load-flow equations introduce the
amount of losses in the generation dispatch. This definition
of the merit order replaces the balancing mechanism, which
is indeed used to compensate for the mismatch of losses in
a lossless merit order. This particular type of optimal power
flow problem is solved using AMPL [11].
C. Transmission loss allocation methods
Based on the generation dispatch PG1 , . . . , PGNG , the
amount of losses PGLi allocated to generator i is assessed
using a TLA method. We study three methods, namely the
“Pro-Rata”, the “Proportional Sharing” and the “Equivalent
Bilateral Exchange” methods, which are detailed hereafter.
1) Pro-Rata (PR): The PR method has been used for
decades in many power systems. Losses are allocated proportionally to the active power injection of every generator
regardless of its location [12]. This TLA method is used for
instance in France, England and Wales [13].
2) Proportional Sharing (PS): The PS method has been
introduced by J. Bialek in [14]. This method is based on power
flow tracing and relies on the assumption that a network node
is a perfect mixer of incoming flows. For each node, every
outcoming active power flow is proportionally composed of
the incoming flows. For each line, the losses are proportionally
allocated to the incoming flows into this line. This method
has been vastly commented and has influenced the design of
many closely related TLA schemes (see, e.g., [15]). No reallife application has yet been reported.
3) Equivalent Bilateral Exchange (EBE): The EBE method
has been proposed by Galiana et al. in [16]. The equivalent
bilateral exchanges are deduced from the application of the PS
principle to the whole network reduced to one node. Losses
are then allocated to those equivalent bilateral exchanges
using incremental transmission loss factors. It is actually
independent of choice of the slack bus [4]. This method has
not yet been implemented.
III. T RANSMISSION LOSS COST PREDICTION
In the heart of our agent-based approach to evaluate the
performances of TLA methods when assuming that the generators are willing to adopt a perfectly competitive behavior,
there is a module that determines how each agent is going to
predict its loss cost. The design adopted for this module is
described on Figure 4. In a few words, with such a choice, a
generator estimates its transmission loss cost (per generated
M W h) by computing a weighted average of its past loss
allocations (per generated M W h). It is clear that, in reallife, power system utilities may rely on other approaches to
estimate accurately those costs. In particular, they may rely on
more sophisticated algorithms or use some specific expertise to
predict those costs. In this respect and by assuming that there
are persons or a group of persons in charge of those prediction
tasks for the utilities, one strategy to design better agents could
have been to analyze their predictions using various types of
data-mining techniques. Agents would then better reflect the
real-life behavior of the utilities.
Since it is very likely that obtaining such prediction data
is going to be difficult (and even sometimes impossible, since
they can not exist for TLA schemes that have not yet been
implemented), another strategy would have been to rely on
experimental economics [17] to generate those data. It consists
of laboratory experiments, where the humans (here playing the
role of people in charge of predicting the loss costs for the
utilities) are repeatedly asked to make decisions in face of
a feedback signal (here the past allocations) related to their
decisions. However, experimental economics is also limited
since there is an obvious bias between the decisions taken
in the laboratory and in real-life due to, for example, the
prior-knowledge a market participant interacting with the real
environment may have on the relevance of its decisions.
IV. P ERFORMANCE EVALUATION OF TLA METHODS
By simulating the influence of TLA methods on an electricity market, we aim to measure the efficiency of the economic
signal they intend to provide.
The market efficiency index adopted in this paper is the social welfare of the system which is the sum of the generators’
surplus and the demands’ surplus [18]. The generator surplus
SPG is defined by:
SPG = M CP ×
ND
X
PDj −
j=1
NG
X
GCGi (PGi )
(3)
i=1
where GCGi (PGi ) is generator i’s total generation costs. As
the demand is fixed, the demand surplus SPD is defined as
follows:
ND
X
SPD = −M CP ×
(4)
PDj
j=1
Consequently, the social welfare of the system can be written:
SP = −
NG
X
GCGi (PGi )
(5)
i=1
The system social welfare will be compared with its optimal
value SP ∗ , which is computed by minimizing the total production costs of the system.
V. S IMULATION RESULTS
After describing the simulation benchmark, this section
reports and analyzes the simulation results.
A. Benchmark
Simulations are run on the 2-bus system that is depicted in
Figure 5. The numerical values of the different parameters of
the system are given in Table I.
When the generators submit an offer equal to their marginal
production cost, one can observe that a large amount of power
is transmitted from bus 1 to bus 2 with a loss rate close to
15%.
The transmission line is modeled by a resistance R1−2 and
an inductance X1−2 . The voltage at each bus is regulated by
its associated generator at 1.0 per unit.
The generation cost supported by generator i can be written
as:
GCGi (PGi ) = aGi × PGi 2 + bGi × PGi + cGi
(6)
L,t
, the quantity of losses allocated to generator i at time t: T LAtGi , its active
Input: The loss cost prediction for time t: ĈG
i
power injection at time t: PGt i and the market price at time t: M CP t .
Parameter: A memory factor β (β ∈ [0, 1]) that weights the actual loss cost at time t−k by a factor β t−k in the average-based
predictions.
L,t+1
.
Output: The loss cost prediction for time t + 1: ĈG
i
Algorithm:
T LAGi ×M CP
L,t
.
• Step 1: Generator i computes the average transmission loss cost per M W h at time t: CG =
PG
i
i
•
L,t+1
=
Step 2: Generator i computes the predicted loss costs for time t + 1 as follows: ĈG
i
Fig. 4.
L,t
L,t
t×β×ĈG
+(t×(1−β)+1)CG
i
i
t+1
.
L,t+1
The procedure a generator i uses at time t to predict its transmission loss cost per generated M W at time t + 1 (ĈG
). At time t = 0, no data
L,0
L,0
for the past loss cost allocations are available and, ĈG
and CG
are chosen equal to 0.
i
G1
G2
1
2
D1
D2
Fig. 5.
Symbol
PD 1
PD 2
QD1
QDA
R1−2
X1−2
a G1
b G1
c G1
a G2
b G2
c G2
2-bus system.
Unit
MW
MW
M V ar
M V ar
p.u.
p.u.
=C/M W h2
=C/M W h
=C
=C/M W h2
=C/M W h
=C
i
i
Value
300
1200
100
100
0.0002
0.0005
0.0147
20
0
0.0588
20
0
TABLE I
N UMERICAL DATA FOR THE 2- BUS SYSTEM .
This leads to an optimal system welfare SP ∗ equal to
−66959=C.
At time t = 0, generators’ estimated loss costs are set to
0=C/M W . In our simulations, a memory factor β has been
chosen equal to 0.5. We have observed that this parameter
does not significantly impact the results, except for the speed
of convergence.
B. Numerical results
1) Market efficiency: Results in terms of system welfare
are represented in Figure 6. One can observe that the pro-rata
allocation induces no change in the generation dispatch. The
system welfare remains equal to −70425=C in this case. As
emphasized in [19], the pro-rata method is poorly efficient in
the context of an asymmetrical system with high losses, as it
is the case here.
On the other hand, EBE and PS methods provide an
economic signal which motivates a change in the generation
dispatch, leading to a greater welfare. After a few iterations
(four to six), the system welfare reaches −68192=C and
−68685=C for the PS and EBE methods, respectively.
When compared to SP ∗ , the TLA methods under consideration may however appear to be inefficient. The lack of
performance might be caused by structural defaults of the
methods themselves (as for the pro-rata method, for example).
It also shows that further research may be needed to design
more efficient allocation schemes. While allocating an amount
of losses larger than the physical losses PL is controversial
[4], one could also think of defining negative allocations for
generators that help decreasing the amount of transmission
losses when they inject more power into the grid. However,
this may potentially lead to volatile markets where predictions
could be less accurate and, therefore, discourage generators to
adopt a perfectly competitive behavior since they may prefer
to cover themselves for the risk of underestimating their loss
costs.
2) Market price: Results in terms of M CP are represented
in Figure 7. Four to six iterations are required before the
market price and the system welfare can be assumed to have
converged. Moreover, simulations run with an initial value of
L,0
set to 20.0=C/M W rather than
the predicted loss costs CG
i
=
0.0 C/M W have shown convergence to the same equilibrium
point.
As one can observe, with the agents chosen to model the
generators, the market clearing price grows with the loss cost
predictions. Except for the first iteration, the M CP is always
larger than the marginal production costs of the generators.
This was expected since, even if the generators adopt a
perfectly competitive behavior, they submit an higher offer
than their marginal production cost to cover the transmission
loss cost they are expecting to be charged for.
3) Discrepancy between the predicted and the actual loss
costs: We have observed that the transmission loss costs
tended to be underestimated by the different generators. This is
mainly due to the fact that the initial estimation of those costs
is set equal to 0. We note however that this under estimation
SP (k=
C)
−68
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
−69
b
b
b
(time-invariant system, simple loss cost prediction algorithms,
2-bus power system, etc), those results suggest that there is still
room for designing more efficient TLA methods. In particular,
we believe that new TLA methods should at least intrinsically
lead to a nearly optimal social welfare when the generators
are willing to adopt a perfectly competitive behavior.
ACKNOWLEDGMENT
−70
b
b
b
b
b
b
b
b
b
b
Damien Ernst is a Research Associate of the Belgian FNRS
from which he acknowledges the financial support.
−71
0
2
4
6
8
10
R EFERENCES
Iteration
Fig. 6. Social welfare at each iteration for the 2-bus system with an averagebased loss cost prediction. The solid, dotted and dashed lines correspond to
the pro-rata, PS and EBE methods, respectively.
MCP (=C)
72
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
70
b
b
b
68
b
b
b
b
b
66
64
62
b
60
0
2
4
6
8
10
Iteration
Fig. 7. Market price M CP at each iteration for the 2-bus system with
an average-based loss cost prediction. The solid, dotted and dashed lines
correspond to the pro-rata, PS and EBE methods, respectively.
vanishes with the iterations.
VI. C ONCLUSION
We have analyzed in this paper the performance of three
transmission loss allocation methods, namely the pro-rata, proportional sharing and equivalent bilateral exchange methods,
for a market in which the power generation companies are
willing to adopt a perfectly competitive behavior. To perform
this analysis, we have relied on an agent-based approach.
As main finding of our analysis, we noted that the three
allocation methods under consideration were leading to a
social welfare which was smaller that the one that could
be obtained by considering an optimized vertically integrated
system. The pro-rata method offered the poorest performance
while the proportional Sharing and the equivalent bilateral
exchange were giving similar performance. While many simplifying assumptions have been adopted in our simulations
[1] A. Conejo, J. Arroyo, N. Alguacil, and A. Guijarro, “Transmission
loss allocation: A comparison of different practical algorithms,” IEEE
Transactions on Power Systems, vol. 17, no. 3, pp. 571–576, August
2002.
[2] F. Galiana, A. Conejo, and I. Kockar, “Incremental transmission loss
allocation under pool dispatch,” IEEE Transactions on Power Systems,
vol. 17, no. 1, pp. 26–32, February 2002.
[3] A. Conejo, F. Galiana, and I. Kockar, “Z-bus loss allocation,” IEEE
Transactions on Power Systems, vol. 16, no. 1, pp. 105–110, February
2001.
[4] J. Mateus and P. Franco, “Transmission loss allocation through equivalent bilateral exchanges and economical analysis,” IEEE Transactions
on Power Systems, vol. 40, no. 4, pp. 1799–1807, November 2005.
[5] T. Peng and K. Tomsovic, “Congestion influence on bidding strategies
in an electricity market,” IEEE Transactions on Power Systems, vol. 18,
no. 3, pp. 1054–1061, August 2003.
[6] E. Bompard, W. Lu, and R. Napoli, “Network constraint impacts on
the competitive electricity markets under supply-side strategic bidding,”
IEEE Transactions on Power Systems, vol. 21, no. 1, pp. 160–170,
February 2006.
[7] A. Rahimi and A. Sheffrin, “Effective market monitoring in deregulated
electricity markets,” IEEE Transactions on Power Systems, vol. 18, no. 2,
pp. 486–493, May 2003.
[8] L. Tesfatsion and K. Judd, Handbook of Computational Economics:
Agent-Based Computational Economics. Elsevier Science and Technology Books, 2006.
[9] S. Stoft, Power System Economics: Designing Markets for Electricity.
IEEE Computer Society Press, 2002.
[10] J. Carpentier, “Contribution à l’étude du dispatching économique,”
Bulletin de la Société Française des Electriciens, vol. 3, pp. 431–447,
1962.
[11] R. Fourer, D. Gay, and B. Kernighan, AMPL: A Modeling Language for
Mathematical Programming. Duxbury Press / Brooks/Cole Publishing
Company, 2002.
[12] F. Galiana and M. Phelan, “Allocation of transmission loss to bilateral
contracts in a competitive environment,” IEEE Transactions on Power
Systems, vol. 15, no. 1, pp. 143–150, February 2000.
[13] J. Bialek, S. Ziemianek, and N. Abi-Samra, “Tracking-based loss allocation and economic dispatch,” in Proceedings of the 13th Power Systems
Computation Conference, Trondheim, Norway, June 1999, pp. 375–381.
[14] J. Bialek, “Tracing the flow of electricity,” in IEE Production, Generation, Transmission and Distribution No. 4, July 1996, pp. 313–320.
[15] J. Bialek and P. Kattuman, “Proportional sharing assumption in tracing
methodology,” IEE Proceedings Genereration, Transmission and Distribution, vol. 151, no. 4, pp. 526–532, July 2004.
[16] F. Galiana, A. Conejo, and H. Gil, “Transmission network cost allocation
based on equivalent bilateral exchanges,” IEEE Transactions on Power
Systems, vol. 18, no. 4, pp. 1425–1431, November 2003.
[17] V. Smith, “Economics in the laboratory,” The Journal of Economic
Perspectives, vol. 8, no. 1, pp. 113–131, May 1994.
[18] A. Mas-Collel, Microeconomic Theory. Whinston and Green, 1995.
[19] Y. Phulpin, M. Hennebel, and S. Plumel, “A market based approach to
evaluate the efficiency of transmission loss allocation,” in Proceedings
of the Modern Electric Power System conference, Wroclaw, Poland,
September 2006, pp. 57–61.
Jing Dai received the B.Eng. degree in control engineering from Beijing
Institute of Technology, China in 2004 and graduated from Supélec, France
in 2007. He is currently pursuing a Ph.D. degree at Supélec.
Yannick Phulpin received in 2004 the M.Sc. degree in electrical engineering
both from Supélec, France and from the Technische Universität Darmstadt,
Germany. He is currently an assistant professor at Supélec and he is pursuing
a Ph.D. in the field of voltage stability and reactive power management at
interconnections under the supervision of Prof. Miroslav Begovic from the
Georgia Institute of Technology in Atlanta, USA.
Vincent Rious received the M.Sc. degree in electrical engineering from
Supélec in France (2004) and the Ph.D. degree in Economics from the University Paris-Sud XI (2007). He is currently an assistant professor at Supélec. His
main topics of research are on experimental economics, investment problem in
the energy markets, and management of interconnectors and their interaction
with massive wind power.
Damien Ernst received the M.Sc. and Ph.D. degrees from the University of
Liège in 1998 and 2003, respectively. He is currently a Research Associate of
the FNRS (Belgian National Fund of Scientific Research) and he is affiliated
with the Systems and Modelling Research Unit of the University of Lige.
Damien Ernst spent the period 2003-2006 with the University of Lige as a
Postdoctoral Researcher of the FNRS and held during this period positions as
visiting researcher at CMU, MIT and ETH. He spent the academic year 20062007 working at SUPELEC (France) as professor. His main research interests
are in the field power system dynamics, optimal control, reinforcement
learning and design of dynamic treatment regimes.